I would go with the cement footing and the "post brackets" they will be plenty strong enough for how you plan to mount the swing. Many folks under the mistaken impression that Pressure treated lumber doesn't rot, but if you look it up, It also says that pressure treated lumber should "not make direct contact with ground"

I'm leaning that way as well. My big concern with mounting a swing would be issues created by movement. I think I'd be fine with the weight, it's the weight in motion that I worried would be a problem. I probably ought to just bag the swing- but it looks so pretty in my mind!

If you check the manufacturer specs on the brackets, you'll see that they are specifically not intended or approved for a free standing structure. They simply do not have the mechanical strength that's needed to comply with codes concerning resistance to lateral forces. They are generally only approved for use with a structure that is attached to something else, like a house. I know a lot of people use them for things like fences and pergolas, and it may seem logical, but it's not ok.

You should have no problem with pressure treated posts in concrete...if they're installed properly. That means that you fill the bottom of the hole with about 4" of gravel for drainage, rest the bottom of the post on top of the gravel, and pour the concrete so that the sides of the post are encased, but not the bottom. This will be a much stronger construction, and allow you to do stuff like have a swing without the structure loosening up or leaning over time.

Thank you Kudzu- that was exactly what I needed to know. I could not think of the term 'lateral forces' but that's exactly what I was thinking of. Since I have to dig some big deep holes anyway, I will go with the posts in ground.

I did read that some people treat the sides of in ground posts with roofing tar (not the bottom?) but I wonder if that would just seal in more water than it keeps out? Depth of frost in my area is about 2-feet 6-inches. I'm thinking I should go a little deeper to improve the strength of the structure(maybe a hole depth of 3-feet 6-inches with 6 inches of gravel in the bottom? Any suggestions on that?

Something I might do is get a 55 gallon barrel full of water and stand the treated wood up in it (maybe leaning against the house) leave it for a week, then put them in the ground and pour concrete right away, wood expands when it gets wet, but concrete does not.

I have a different take on this--how difficult will it be to replace those posts, and they will surely deteriorate, after they have been cee-mented into the ground? Our pergola posts (and indeed porch support posts) sit on-top of bolts (or rebar, I forget) drilled-into the concrete.

So another way would be: pour a footing, insert-or-drill-and-install a bolt or rebar piece sticking-up 4-6", drill a receiving hole into the bottom of your post, and place it. It's not going anywhere in re: the swing...

Whether this idea works for you or not may depend on where your structure is getting its lateral bracing. If this depends on your posts being "rigidly upright" then I guess you gotta bury them.